Finite Group Databasebeta

← #77263

Subgroups — #77263

Order 384.

592 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#946C212yesmaximal, complemented
2C2×#896C212yesmaximal, complemented
3C3×#395C211yesmaximal, characteristic, complemented
4#4994C212yesmaximal, complemented
5#4988C211yesmaximal, characteristic
6#4979C211yesmaximal, characteristic, complemented
7#4676C212yesmaximal, complemented
8#4665C212yesmaximal
9#4647C212yesmaximal, complemented
10#2683—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×D12C2211yescharacteristic
13C22×C4×D3C2212yes—
14C22×Dic6C2211yescharacteristic
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17#946—28no—
18C2×(C2×D3)⋊C4C2212yes—
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×C4⋊Dic3C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29#896—28no—
30C3×#95C2212yes—
31C3×#95C2211yescharacteristic
32C3⋊#95C2212yes—
33C3⋊#95C2211yescharacteristic
34C2×C22⋊Q8—32no—
35C2×C22⋊Q8—32no—
36#396—32no—
37#396—32no—
38#395—31no—
39#395D311yescharacteristic, complemented
40#390—31no—
41#389—32no—
42#386—32no—
43C23×D3C2312yes—
44C2×C3×Q8—24no—
45C2×C3×Q8—41no—
46C22×C3×C4C2311yescharacteristic
47C22×C3×C4C2312yes—
48C22×C3×C4C2311yescharacteristic
49C22×C3×C4C2312yes—
50C22×C3×C4C2311yescharacteristic
51C22×Dic3C2312yes—
52C22×Dic3C2312yes—
53C22×Dic3C2312yes—
54C2×D12—41no—
55C2×D12—41no—
56C2×D12Q814yescomplemented
57C2×C4×D3—24no—
58C2×C4×D3—24no—
59C2×C4×D3—42no—
60C2×Dic6—24no—
61C2×Dic6—41no—
62C3×C4⋊C4—24no—
63C3×C4⋊C4—41no—
64(C2×D3)⋊C4—24no—
65(C2×D3)⋊C4—24no—
66(C2×D3)⋊C4—24no—
67(C2×D3)⋊C4—24no—
68(C2×D3)⋊C4—42no—
69C4⋊Dic3—24no—
70Dic3⋊C4—24no—
71Dic3⋊C4—24no—
72Dic3⋊C4—42no—
73Dic3⋊C4D418yescomplemented
74C4×Dic3—41no—
75C22×Q8—31no—
76C22×Q8C2×D311yescharacteristic
77C22×D4—31no—
78C23×C4—32no—
79C22⋊Q8—68no—
80C22⋊Q8—68no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4C2×D312yes—
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C22⋊C4—32no—
89C2×C22⋊C4—32no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C42—31no—
94#95—32no—
95#95—31no—
96#95C2×D312yes—
97#95C2×D311yescharacteristic
98C23×C3C2411yescharacteristic
99C22×D3—22no—
100C22×D3—22no—
101C22×D3—22no—
102C22×D3—22no—
103C22×D3—22no—
104C22×D3—22no—
105C22×D3C2×Q814yes—
106C3×Q8—44no—
107C2×C3×C4—22no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4C2×D414yes—
119C2×C3×C4C2×Q812yes—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3C2×D414yes—
128C2×Dic3C2×D414yes—
129D12—44no—
130C4×D3—48no—
131Dic6—44no—
132C24—32no—
133C2×Q8—24no—
134C2×Q8—41no—
135C2×Q8—64no—
136C2×Q8—121no—
137C2×D4—34no—
138C2×D4—121no—
139C2×D4—121no—
140C22×C4—32no—
141C22×C4—32no—
142C22×C4—32no—
143C22×C4—64no—
144C22×C4—64no—
145C22×C4—122no—
146C22×C4C22×D311yescharacteristic
147C22×C4C22×D312yes—
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D312yes—
150C22×C4C22×D311yescharacteristic
151C4⋊C4—24no—
152C4⋊C4—38no—
153C4⋊C4—41no—
154C4⋊C4—64no—
155C4⋊C4—64no—
156C4⋊C4—64no—
157C4⋊C4—122no—
158C22⋊C4—64no—
159C22⋊C4—64no—
160C22⋊C4—64no—
161C22⋊C4—64no—
162C22⋊C4—122no—
163C42—121no—
164C22×C3C22×D412yes—
165C22×C3C22×Q811yescharacteristic
166C22×C3H(5,𝔽2)11yescharacteristic
167C22×C3H(5,𝔽2)11yescharacteristic
168C22×C32−1+412yes—
169C2×D3—24no—
170C2×D3—24no—
171C2×D3—24no—
172C2×D3—42no—
173C2×D3—42no—
174C2×D3—42no—
175C2×D3—42no—
176C3×C4—22no—
177C3×C4—24no—
178C3×C4—42no—
179C3×C4—41no—
180C3×C4—41no—
181Dic3—24no—
182Dic3—24no—
183Dic3—42no—
184C23—34no—
185C23—62no—
186C23—62no—
187C23—62no—
188C23—62no—
189C23—62no—
190C23—62no—
191C23C23×D311yescharacteristic
192Q8—44no—
193Q8—124no—
194D4—124no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—34no—
207C2×C4—34no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—128no—
216C2×C4D3×D414yes—
217C2×C4D3×Q812yes—
218C2×C3#53712yes—
219C2×C3#54311yescharacteristic
220C2×C3D4211yescharacteristic
221C2×C3D4×Q812yes—
222C2×C3#56311yescharacteristic
223D3—44no—
224C22—64no—
225C22—64no—
226C22—64no—
227C22—122no—
228C22—122no—
229C22—122no—
230C22—122no—
231C22C2×D3×D412yes—
232C22#100411yescharacteristic
233C22C2×D3×Q811yescharacteristic
234C22#100712yes—
235C22#100911yescharacteristic
236C4—22no—
237C4—24no—
238C4—41no—
239C4—41no—
240C4—42no—
241C4—64no—
242C4—64no—
243C4—122no—
244C3#268311yescharacteristic, complemented, Sylow
245C2—124no—
246C2#559011yescharacteristic
247C2#562912yes—
248C2#563411yescharacteristic
249C2#572712yes—
250C2#573511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC22×D12C22
hypercenter#395D3
hypocenterC3#2683
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#946C2semidirect
2. C2×#896C2semidirect
3. C3×#395C2semidirect
4. #4994C2semidirect
6. #4979C2semidirect
7. #4676C2semidirect
9. #4647C2semidirect
39. #395D3semidirect
56. C2×D12Q8semidirect
73. Dic3⋊C4D4semidirect
244. C3#2683semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#9462
C2×#8962
C3×#3952
#49942
#49882
#49792
#46762
#46652
#46472
#26833